Big Picture Semantics how do we figure out the situations in which - - PowerPoint PPT Presentation

Big Picture

Semantics – how do we figure out the situations in which sentences are true or false? Compositionality = pieces + composing them

• Lexical semantics = what do we know about

word meanings?

• Compositional semantics = how do we put

the pieces together?

Compositional semantics

So far:

• Sentence-meaning
• truth, truth-conditions, possible worlds
• Meaning of NPs (noun phrases)
• constant/variable reference, naming game
• Meaning of predicates

(verbs,nouns,adjectives)

• set theory, relations, functions
• Putting things together
• lambdas, types

Compositional semantics: NPs

From now on: More about NPs & predicates Keep comparing theory & data!

• NPs that don’t refer to objects

sets of sets, patterns of meaning: polarity

• Different types of NPs, & what they do

definite, indefinite, quantificational

• A unified theory

kinds, objects, mass, count, different languages

To motivate further theory

Either John is in that room or Mary is, and possibly they both are.

• What are some problems in translating this into

predicate logic? Stuff we need: “that” (to make “that room”) representing “either...or” “possibly

To motivate further theory

(1) Either John is in that room or Mary is, and possibly they both are.

• What are some problems in translating this into

predicate logic? Stuff we need: “that” (to make “that room”) representing “either...or” “possibly”

(2) Sam wants a dog, but Alice wants cats (3) A dog is a quadruped

• What are some problems with translating this?

Stuff we need: plural vs singular phrases what to do with bare plurals? Is “a dog” ambiguous? “but” vs. “and” representing generic meanings New kind of ambiguity?

(2) Sam wants a dog, but Alice wants cats (3) A dog is a quadruped

• What are some problems with translating this?

Stuff we need: plural vs singular phrases what to do with bare plurals? Is “a dog” ambiguous? “but” vs. “and” representing generic meanings New kind of ambiguity?

Scope ambiguity

Lexical or structural?

• Sam wants a dog
• Everything is black or white
• Someone loves everyone

Semantic theory so far:

• Sentence = predicate saturated with all its

arguments (so, smth True or False)

• Sentences can be composed from other

sentences using “no”, “and”, “or”,”if-then”

• Predicates can have arity=valency of zero (to

rain), one (to run), two (to devour), three (to give) etc. arguments.

• Arguments can be of any type, including

entities, other predicates, and whole sentences

Semantic theory so far (cont’d):

• NPs can represent entities (John), predicates

(“a dog” in “Fido is a dog”), or expressions with quantifiers (“a dog” in “Sam wants a dog”)

• We can make new predicate expressions using

lambdas; also new semantic rules.

Semantic rules

Lambda abstraction

• Used when something moves

John λx I like x = I like John

• Used for making relative clauses & questions

Who λx x does it = set of people who do it

• Used for representing predicates

Not smoking is healthy = Healthy (λx ~smoke(x))

Semantic rules (cont’d)

Function application:

• Used to put predicates and arguments together

John runs John λx I like x Someone runs Conjunction and other ‘connective’ rules:

• Take predicates that you want to conjoin
• Fully saturate them using variables
• Conjoin the resulting sentences
• Lambda abstract over the variables to get the

new predicate of correct type

Generalised Quantifiers

• Try applying conjunction schema to

“John and Mary”

• What is the type of these expressions?

“Every guy but John” “Some apples and this pear”

• Even worse: one might initially think that

a unicorn is referential (refers to a particular individual) e.g., A unicorn was there. He was beautiful.

• However, as Bertrand Russel noted,

indefinites are also non-referential: Nobody has seen a unicorn, because there aren't any.

Generalised Quantifiers (cont'd)

Generalised Quantifier Theory

• Basic idea:

All NPs are of the same type – each is a set of sets. [[Jane]] = {snore, run, talk, girl}

Generalised Quantifier Theory

• Basic idea:

All NPs are of the same type – each is a set of sets. [[Jane]] = {snore, run, talk, girl} λP.P(j)

Generalised Quantifier Theory

• Basic idea:

All NPs are of the same type – each is a set of sets. [[Jane]] = {snore, run, talk, girl} λP.P(j) (e → t) → t

Syntax and Semantics

S N P V P [[Jane snores]] = λP.P(j) (snore) = (e→t)→t e → t

Syntax and Semantics

S N P V P [[Jane snores]] = λP.P(j) (snore) = snore (j) (e→t)→t e → t t

Syntax and Semantics

Mismatch for syntax and semantics:

• What's the argument?
• What's the predicate?
• What is the constituent structure?
• Which individuals matter for the truth of S?

[[Every student danced]] = Every x [ student(x) → danced(x) ]

Syntax and Semantics

In English:

• “Every student” is a unit
• It combines with “danced”

In PC:

a totally different tree

• [

[ every student ] ] and [ [danced] ] are not constituents!

• “student danced” is a unit
• It combines with “every”

Syntax and Semantics

In English:

• Look in the set of students

– If all members of this

set danced – T

– If not all members of

this set danced – F

In PC:

• Look at all the entities in the

universe

– If the entity is not a student,

T

– If the entity is a student,

then if this entity danced – T

• therwise - F

GQ Theory

• Try semantics which is more true to syntax:

[[Every student]] = {dance, run, talk, student} λP.Every(student)(P) (e→t)→t What’s “every”? Something that combines with “student” to make “every student”

GQ Theory

• What a determiner might mean:

“every” - something that combines with “student” e → t to make “every student” (e→t) → t

GQ Theory

• What a determiner might mean:

“every” - something that combines with “student” e→t λQe→t. to make “every student” λPe→t.Every(student)(P)

GQ Theory

• “every” - combines with

“student” e → t λQe→t. to make “every student” λPe→t.Every(student)(P)

• SO: [[Every]] =

λQλP.Every(Q)(P) (e→t)→((e→t)→t)

GQ Theory

• Try semantics which is more true to syntax:

S N P V P Det N ' V [[Every student danced]] = (e→t)→((e→t)→t e→t e→t

GQ Theory

• Try semantics which is more true to syntax:

S N P V P Det N ' V [[Every student danced]] = (e→t)→((e→t)→t e→t e→t

λQλP.Every(Q)(P) student' danced'

GQ Theory

• Try semantics which is more true to syntax:

S N P V P Det N ' V [[Every student danced]] = (e→t)→((e→t)→t e→t e→t

λQλP.Every(Q)(P) student' danced'

GQ Theory

• Try semantics which is more true to syntax:

S N P V P Det N ' V [[Every student danced]] = (e→t)→((e→t)→t e→t e→t

λQλP.Every(Q)(P) student' danced' λQλP.∀x(Q(x)→P(x)) λx.student'(x) λy.danced'(y)

Several GQs

Generalised Quantifier Theory

Several GQs Several Determiners

• [

[All NP] ]= [ [All (A,B)] ]=

• [

[Some NP] ]= [ [Some (A,B)] ]=

• [

[No NP] ]= [ [No (A,B)] ]=

• [

[At least 5 NP] ]= [ [At least 5 (A,B)] ]=

• [

[Most NP] ]= [ [Most (A,B)] ]=

Generalised Quantifier Theory

Several GQs Several Determiners

• [

[All NP] ]= [ [All (A,B)] ]= All Ling 130 students are smart .

Generalised Quantifier Theory

Several GQs Several Determiners

• [

[All NP] ]= [ [All (A,B)] ]=

• [

[Some NP] ]= [ [Some (A,B)] ]= Some Brandeis students commute.

Generalised Quantifier Theory

Several GQs Several Determiners

• [

[All NP] ]= [ [All (A,B)] ]=

• [

[Some NP] ]= [ [Some (A,B)] ]=

• [

[No NP] ]= [ [No (A,B)] ]= No boy(s) came to the party.

Generalised Quantifier Theory

Several GQs Several Determiners

• [

[All NP] ]= [ [All (A,B)] ]=

• [

[Some NP] ]= [ [Some (A,B)] ]=

• [

[No NP] ]= [ [No (A,B)] ]=

• [

[At least 5 NP] ]= [ [At least 5 (A,B)] ]= At least 5 ballerinas danced there.

Generalised Quantifier Theory

Several GQs Several Determiners

• [

[All NP] ]= [ [All (A,B)] ]=

• [

[Some NP] ]= [ [Some (A,B)] ]=

• [

[No NP] ]= [ [No (A,B)] ]=

• [

[At least 5 NP] ]= [ [At least 5 (A,B)] ]=

• [

[Most NP] ]= [ [Most (A,B)] ]= Most Brandeis students live on campus.

Generalised Quantifier Theory

Several GQs Several Determiners

• [

[All NP] ]= [ [All (A,B)] ]=

• [

[Some NP] ]= [ [Some (A,B)] ]=

• [

[No NP] ]= [ [No (A,B)] ]=

• [

[At least 5 NP] ]= [ [At least 5 (A,B)] ]=

• [

[Most NP] ]=

[

[Most (A,B)] ]=